## Author: Dr. Jonathan Kenigson, FRSA

The f (R) metric stability of an arbitrary black hole is not a purely metric question. Field couplings complicate the picture dramatically. Einstein’s equation states four consistent f (R) gravitational conditions involving R-derivatives in the Kerr and Kerr-Newman cases. To correct this difficulty, we convert f (R) gravity into a combined scalar-tensor theory. In this regard, a continuous curvature scalar is a “Scalaron,” indicating that all line numbers are equal and consequently forming a generalization of Brans-Dicke Theory. It is difficult but possible to prove the case that the black hole of type f (R) is stable when the Scalaron has no Tachyonic dimensions. Indeed, f(R) gravity [1, 2, 3] has received scholarly focus because it makes novel predictions regarding the expansion and acceleration of the universe [4]. With an additional scalar in Einstein’s Equation, f(R) Gravitons are named as “Einstein Gravitons,” which are also known as massless Gravitons. As an example, researchers have demonstrated that metric-f(R) gravity is equivalent equal to $w-BD=0$ where BD is Brans-Dicke [5]. Although the Equivalence Principle as tested by solar-system constraints places *a-posteriori* conditions on f(R) gravities, these theories cannot be spontaneously excluded if the Chameleon Mechanism is used to solve the constraints on the generalized paradigm in f (R, T). It has been shown that Equivalence Principle testing enables models of f (R) that cannot differentiate from the ΛCDM model to determine initial inflationary conditions for universes [6]. However, it is not necessary to specify strong Isotropy conditions in the sense of the article cited [7].

To be acceptable as a gravitational model, f(R)-metrics must meet specific minimum prerequisites for the theoretical and observational feasibility [2,5]. Three main physical/mathematical requirements are a correct metric cosmological dynamic, freedom from instabilities (tachyons) and Fadeev-Popov Ghosts [8, 9, 6], and convergence to the correct Newtonian and post-Newtonian limits. The Schwarzschild-de-Sitter black hole f(R) solution for positive constant curvature scalar is furnished in [7] and other black hole solutions have been found for a scalar-tensor gravity with non-constant curvature [10]. Closed-form solutions can employ high-dimensional Hyperbolic geometry: Structural equations are secured from f(R) gravity by considering the negative constant curvature as mathematically equivalent to the Schwarzschild-AdS (SAdS) black hole. To get the constant-curvature black hole solution of f(R) gravity coupled to matter, the Stress-Energy Tensor must vanish. Two very familiar field couplings are to the Maxwell [12] and Yang Mills fields [13].

Black hole solutions must pass stability (as well as feasibility) tests. There are two ways to achieve black hole stability: First, complete stability arises from a Group-Theoretic perspective by taking into account even and odd (symmetric parity) perturbations [14] and, on the other hand, spherically symmetric perturbations for simplification using Schwarzschild metrics [15]. The second condition is not sufficient to ensure complete BH stability and must therefore be augmented by thermodynamic (heat capacity) conditions [16]. The main idea is to manage to get second-order inhomogeneous Schrödinger-type equations for the physical fields present in the referenced article. The black hole is mathematically stable when all potentials are positive outside the event horizon (in the absence of postulated Hawking radiation). Proving the stability of the Kerr black hole is difficult because Kerr stability conditions in f(R) gravity consist of fourth order derivatives in the Einstein equations [17,18].

Works Cited/Further Reading.

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[2]. B.P. Abbott, R. Abbott, T.D. Abbott, M.R. Abernathy et al., Phys. Rev. Lett. 116, 061102 (2016)

[3]. A.G. Riess et al., Astron. J. 116, 1009 (1998).

[4]. S. Perlmutter et al., Astrophys. J. 517, 565 (1999).

[5]. R.A. Knop et al., Astrophys. J. 598, 102 (2003).

[6]. R. Amanullah et al., Astrophys. J. 716, 712 (2010).

[7]. D.H. Weinberg, M.J. Mortonson, D.J. Eisenstein, C. Hirata, A.G. Riess, E. Rozo, Phys. Rep. 530, 87 (2013).

[8]. P.J.E. Peebles, B. Ratra, Rev. Mod. Phys. 75, 559 (2003).

[9]. T. Padmanabhan, Phys. Rep. 380, 235 (2003).

[10]. J.M. Overduin, P.S. Wesson, Phys. Rep. 402, 267 (2004).

[11]. H. Baer, K.-Y. Choi, J.E. Kim, L. Roszkowski, Phys. Rep. 555, 1 (2015).

[12]. R. Caldwell, R. Dave, P.J. Steinhardt, Phys. Rev. Lett. 80, 1582 (1998).

[13]. S. Tsujikawa, Class. Quantum Gravity 30, 214003 (2013).

[14]. R.R. Caldwell, Phys. Lett. B. 545, 23 (2002).

[15]. T. Harko, F.S.N. Lobo, S. Nojiri, S.D. Odintsov, Phys. Rev. D 84, 024020 (2011).

[16]. T. Harko, F.S.N. Lobo, Galaxies 2, 410 (2014).

[17]. M. Bojowald, Quantum Cosmology: A Fundamental Description of the Universe (Springer, New York, 2011).

[18]. M. Bojowald, Rep. Progress Phys. 78, 023901 (2015).

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